NOC:Commutative Algebra


Lecture 1 - Zariski Topology and K-Spectrum


Lecture 2 - Algebraic Varieties and Classical Nullstelensatz


Lecture 3 - Motivation for Krulls Dimension


Lecture 4 - Chevalleys dimension


Lecture 5 - Associated Prime Ideals of a Module


Lecture 6 - Support of a Module


Lecture 7 - Primary Decomposition


Lecture 8 - Primary Decomposition (Continued...)


Lecture 9 - Uniqueness of Primary Decomposition


Lecture 10 - Modules of Finite Length


Lecture 11 - Modules of Finite Length (Continued...)


Lecture 12 - Introduction to Krull’s Dimension


Lecture 13 - Noether Normalization Lemma (Classical Version)


Lecture 14 - Consequences of Noether Normalization Lemma


Lecture 15 - Nil Radical and Jacobson Radical of Finite type Algebras over a Field and digression of Integral Extension


Lecture 16 - Nagata’s version of NNL


Lecture 17 - Dimensions of Polynomial ring over Noetherian rings


Lecture 18 - Dimension of Polynomial Algebra over arbitrary Rings


Lecture 19 - Dimension Inequalities


Lecture 20 - Hilbert’s Nullstelensatz


Lecture 21 - Computational rules for Poincaré Series


Lecture 22 - Graded Rings, Modules and Poincaré Series


Lecture 23 - Hilbert-Samuel Polynomials


Lecture 24 - Hilbert-Samuel Polynomials (Continued...)


Lecture 25 - Numerical Function of polynomial type


Lecture 26 - Hilbert-Samuel Polynomial of a Local ring


Lecture 27 - Filtration on a Module


Lecture 28 - Artin-Rees Lemma


Lecture 29 - Dimension Theorem


Lecture 30 - Dimension Theorem (Continued...)


Lecture 31 - Consequences of Dimension Theorem


Lecture 32 - Generalized Krull’s Principal Ideal Theorem


Lecture 33 - Second proof of Krull’s Principal Ideal Theorem


Lecture 34 - The Spec Functor


Lecture 35 - Prime ideals in Polynomial rings


Lecture 36 - Characterization of Equidimensional Affine Algebra


Lecture 37 - Connection between Regular local rings and associated graded rings


Lecture 38 - Statement of the Jacobian Criterion for Regularity


Lecture 39 - Hilbert function for Affine Algebra


Lecture 40 - Hilbert Serre Theorem


Lecture 41 - Jacobian Matrix and its Rank


Lecture 42 - Jacobian Matrix and its Rank (Continued...)


Lecture 43 - Proof of Jacobian Critrerion


Lecture 44 - Proof of Jacobian Critrerion (Continued...)


Lecture 45 - Preparation for Homological Dimension


Lecture 46 - Complexes of Modules and Homology


Lecture 47 - Projective Modules


Lecture 48 - Homological Dimension and Projective module


Lecture 49 - Global Dimension


Lecture 50 - Homological characterization of Regular Local Rings (RLR)


Lecture 51 - Homological characterization of Regular Local Rings (Continued...)


Lecture 52 - Homological Characterization of Regular Local Rings (Continued...)


Lecture 53 - Regular Local Rings are UFD


Lecture 54 - RLR-Prime ideals of height 1


Lecture 55 - Discrete Valuation Ring


Lecture 56 - Discrete Valuation Ring (Continued...)


Lecture 57 - Dedekind Domains


Lecture 58 - Fractionary Ideals and Dedekind Domains


Lecture 59 - Characterization of Dedekind Domain


Lecture 60 - Dedekind Domains and prime factorization of ideals